Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces

Profeta, Angelo; Sturm, Karl-Theodor

doi:10.1007/s00526-020-01774-w

“…Property (i) instead directly follows from the Boundary volume rigidity Theorems 8. Let now ( X , d, H N ) be the doubling of (X, d, H N ) gluing along ∂ X , see for instance [79] for the precise definition. We claim that it is noncollapsed Ricci limit (of a sequence of smooth N -dimensional Riemannian manifolds with no boundary and Ricci curvature bounded from below by −N ).…”

Section: Topological Regularity Up To the Boundarymentioning

confidence: 99%

Boundary regularity and stability for spaces with Ricci bounded below

Bruè

¹

,

Naber

²

,

Semola

³

2022

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This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X.

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“…We extend next the Wasserstein distance to signed measures. We adopt a similar idea to what was suggested in [23,29,15] using a decomposition into positive and negative parts, x (s) = x (s)+ − x (s)− where x (s)+ = max(x (s)+ , 0) and x (s)− = max(−x (s)+ , 0). For any vectors a, b ∈ R p , we define the generalized Wasserstein distance as:…”

Section: Minimum Wassertein Estimatesmentioning

confidence: 99%

Group Level MEG/EEG Source Imaging via Optimal Transport: Minimum Wasserstein Estimates

Janati

¹

,

Bazeille

²

,

Thirion

³

et al. 2019

Lecture Notes in Computer Science

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Magnetoencephalography (MEG) and electroencephalography (EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity. Inferring the location of the current sources that generated these magnetic fields is an ill-posed inverse problem known as source imaging. When considering a group study, a baseline approach consists in carrying out the estimation of these sources independently for each subject. The ill-posedness of each problem is typically addressed using sparsity promoting regularizations. A straightforward way to define a common pattern for these sources is then to average them. A more advanced alternative relies on a joint localization of sources for all subjects taken together, by enforcing some similarity across all estimated sources. An important advantage of this approach is that it consists in a single estimation in which all measurements are pooled together, making the inverse problem better posed. Such a joint estimation poses however a few challenges, notably the selection of a valid regularizer that can quantify such spatial similarities. We propose in this work a new procedure that can do so while taking into account the geometrical structure of the cortex. We call this procedure Minimum Wasserstein Estimates (MWE). The benefits of this model are twofold. First, joint inference allows to pool together the data of different brain geometries, accumulating more spatial information. Second, MWE are defined through Optimal Transport (OT) metrics which provide a tool to model spatial proximity between cortical sources of different subjects, hence not enforcing identical source location in the group. These benefits allow MWE to be more accurate than standard MEG source localization techniques. To support these claims, we perform source localization on realistic MEG simulations based on forward operators derived from MRI scans. On a visual task dataset, we demonstrate how MWE infer neural patterns similar to functional Magnetic Resonance Imaging (fMRI) maps.

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“…Finally, as a M/EEG source estimates can be positive or negative, we extend the Wasserstein distance W u to signed measures. We adopt a similar idea to what was suggested in [32,42,53] using a decomposition into positive and negative parts, a = a + − a − where a + = max(a, 0) and a − = max(−a, 0). For any vectors a, b ∈ R p , we define the generalized Wasserstein distance as:…”

Section: Optimal Transport Backgroundmentioning

confidence: 99%

Multi-subject MEG/EEG source imaging with sparse multi-task regression

Janati

¹

,

Bazeille

²

,

Thirion

³

et al. 2019

Preprint

0

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Magnetoencephalography and electroencephalography (M/EEG) are non-invasive modalities that measure the weak electromagnetic fields generated by neural activity. Estimating the location and magnitude of the current sources that generated these electromagnetic fields is a challenging ill-posed regression problem known as source imaging. When considering a group study, a common approach consists in carrying out the regression tasks independently for each subject. An alternative is to jointly localize sources for all subjects taken together, while enforcing some similarity between them. By pooling all measurements in a single multi-task regression, one makes the problem better posed, offering the ability to identify more sources and with greater precision. The Minimum Wasserstein Estimates (MWE) promotes focal activations that do not perfectly overlap for all subjects, thanks to a regularizer based on Optimal Transport (OT) metrics. MWE promotes spatial proximity on the cortical mantel while coping with the varying noise levels across subjects. On realistic simulations, MWE decreases the localization error by up to 4 mm per source compared to individual solutions. Experiments on the Cam-CAN dataset show a considerable improvement in spatial specificity in population imaging. Our analysis of a multimodal dataset shows how multi-subject source localization closes the gap between MEG and fMRI for brain mapping.

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Heat flow with Dirichlet boundary conditions via optimal transport and gluing of metric measure spaces

Cited by 8 publications

References 14 publications

Boundary regularity and stability for spaces with Ricci bounded below

Boundary regularity and stability for spaces with Ricci bounded below

Group Level MEG/EEG Source Imaging via Optimal Transport: Minimum Wasserstein Estimates

Multi-subject MEG/EEG source imaging with sparse multi-task regression

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